Centering Koebe polyhedra via Möbius transformations

Abstract

A variant of the Circle Packing Theorem states that the combinatorial class of any convex polyhedron contains elements midscribed to the unit sphere centered at the origin, and that these representatives are unique up to Möbius transformations of the sphere. Motivated by this result, various papers investigate the problem of centering spherical configurations under Möbius transformations. In particular, Springborn proved that for any discrete point set on the sphere there is a Möbius transformation that maps it into a set whose barycenter is the origin, which implies that the combinatorial class of any convex polyhedron contains an element midsribed to a sphere with the additional property that the barycenter of the points of tangency is the center of the sphere. This result was strengthened by Baden, Krane and Kazhdan who showed that the same idea works for any reasonably nice measure defined on the sphere. The aim of the paper is to show that Springborn's statement remains true if we replace the barycenter of the tangency points by many other polyhedron centers. The proof is based on the investigation of the topological properties of the integral curves of certain vector fields defined in hyperbolic space. We also show that most centers of Koebe polyhedra cannot be obtained as the center of a suitable measure defined on the sphere.

From spherical to Euclidean illumination

“…To prove the theorem, we adopt some ideas from the proof of Theorem 3 in [9]. Let P be a Koebe polyhedron, i.e., a convex polyhedron in E 3 whose edges are tangent to S 2 .…”

“…Let P be a Koebe polyhedron, i.e., a convex polyhedron in E 3 whose edges are tangent to S 2 . Then there are two families of circles on S 2 associated to P [5,9]. The elements of the first family, called face circles are the incircles of the faces of P; each such circle touches the edges of a face of P at the points where the edges touch S 2 .…”